Binary operations
You are probably familiar with some binary operations, for example, addition and multiplication, but we are going to look at binary operations in more depth.
The definition of a binary operation
A binary operation is simply a function that takes two input values and outputs one value. More precisely, it takes an ordered pair (known as an operand) from the Cartesian product of two sets and produces an element in another set. Using our notation:
An operation can be anything! For example, sexual reproduction within the set of mammals can be considered a binary operation. It takes an ordered pair from the subsets of males and females and produces another member of the set of mammals. More formally:
Within the number systems, addition is a good example of a binary operation. Let's define it for the real numbers:
You'll notice that with binary operations, we don't use the usual function notation of f(x, y), but instead, we use...
